利用未标记数据在多实例学习问题中，以改善在自由生活条件下对帕金森病震颤的检测

Currently, most SSL approaches work in the single-instance learning setting, where the goal is to predict the label $y$$y$yy of a data sample x. However, a different setting with particular interest for remote disease screening is that of MultipleInstance Learning (MIL), where the goal is to predict a label $y$$y$yy for a bag of samples (or instances) $\{{\mathbf{x}}_1,{\mathbf{x}}_2,\dots \}$$\left\{{\mathbf{x}}_1,{\mathbf{x}}_2,\dots \right\}${x_(1),x_(2),dots}\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots\right\}. Throughout the learning procedure, the labels of the instances in the bag are unknown and the only available annotation is a label that describes the entire bag. This situation is regularly encountered in practice. In our case, for example, PD tremor may manifest only for a fraction of time during everyday life, depending on the disease’s stage, the symptom’s intermittence or levodopa intake. Hence, to detect tremor from sensor measurements obtained in-the-wild, we must resort to MIL, as there is no easy way to obtain detailed ground-truth of the on-off periods. 

半监督学习（SSL）是一种情况，除了一个完全标记的集合 ${\mathcal{D}}_l={\left\{({\mathbf{x}}_i,y_i)\right\}}_{i=1}^L$${\mathcal{D}}_l={\left\{\left({\mathbf{x}}_i,y_i\right)\right\}}_{i=1}^L$D_(l)={(x_(i),y_(i))}_(i=1)^(L)\mathcal{D}_{l}=\left\{\left(\mathbf{x}_{i}, y_{i}\right)\right\}_{i=1}^{L} 外，我们还得到了一个从相同边缘分布独立同分布抽取的未标记数据点集合 ${\mathcal{D}}_u={\left\{{\mathbf{x}}_i\right\}}_{i=L+1}^{L+U}$${\mathcal{D}}_u={\left\{{\mathbf{x}}_i\right\}}_{i=L+1}^{L+U}$D_(u)={x_(i)}_(i=L+1)^(L+U)\mathcal{D}_{u}=\left\{\mathbf{x}_{i}\right\}_{i=L+1}^{L+U} 。目标是利用 ${\mathcal{D}}_u$${\mathcal{D}}_u$D_(u)\mathcal{D}_{u} 来学习一个比
what would be possible using only ${\mathcal{D}}_l$${\mathcal{D}}_l$D_(l)\mathcal{D}_{l}. In general, it is not evident how unlabelled data can help, as knowledge of the marginal does not directly contribute to the data likelihood for a given model. In fact, ${\mathcal{D}}_u$${\mathcal{D}}_u$D_(u)\mathcal{D}_{u} can be helpful only if certain assumptions are true. The most common one is the smoothness assumption, which states that if two points ${\mathbf{x}}_1,{\mathbf{x}}_2$${\mathbf{x}}_1,{\mathbf{x}}_2$x_(1),x_(2)\mathbf{x}_{1}, \mathbf{x}_{2} in a highdensity region are close, then so should be their predicted labels $y_1,y_2$$y_1,y_2$y_(1),y_(2)y_{1}, y_{2}. This suggests that the learnt classifier must be smooth in high-density regions. The well-known low-density separation assumption that requires the decision boundary to lie in a low-density region, is an alternative view of the smoothness assumption. 

Early SSL techniques for neural networks were designed to enforce the low-density separation assumption on the resulting classifier. They did so by penalizing a decision boundary in high-density regions, for example by encouraging the model output distribution to have low entropy [38]. More recent techniques employ a similar regularization scheme, in which the model is encouraged to be invariant across label-preserving transformations of the input data (e.g. a small amount of additive gaussian noise that does not change the label). This principle is called consistency regularization and is used in many state-of-the-art methods for semi-supervised image classification, like Pseudo-Ensembles [39], Temporal Ensembling [40] and Mean Teacher [41]. 

一种在本文中特别感兴趣的一致性正则化的方法是虚拟对抗训练（VAT）方法[42]。VAT 不是随机扰动 x，而是计算导致模型输出最大变化的扰动。为了实现这一点，在每一步训练中，它解决以下优化问题：

where $p(y\mid \mathbf{x};\hat{\theta })$$p(y\mid \mathbf{x};\widehat{\theta })$p(y∣x; hat(theta))p(y \mid \mathbf{x} ; \hat{\theta}) is our model, $D$$D$DD is a distribution divergence metric and $\hat{\theta }$$\widehat{\theta }$hat(theta)\hat{\theta} denotes the model parameters at the current step. The optimization problem of Eq. 1 can be approximated efficiently with just an additional forward-backward pass through the network. Having estimated ${\mathbf{r}}_{vadv}$${\mathbf{r}}_{vadv}$r_(vadv)\mathbf{r}_{v a d v}, the model is then encouraged to be smooth along its direction by minimizing the Local Distributional Smoothing (LDS) loss at each data point: 

在多实例学习（MIL）中，我们再次面对一组样本及其标签 $D={\left\{(X_i,y_i)\right\}}_{i=1}^L$$D={\left\{\left(X_i,y_i\right)\right\}}_{i=1}^L$D={(X_(i),y_(i))}_(i=1)^(L)D=\left\{\left(X_{i}, y_{i}\right)\right\}_{i=1}^{L} 。不同的是，这里的每个样本本身就是一个实例包。
i.e. $X_i=\{{\mathbf{x}}_i^1,{\mathbf{x}}_i^2,\dots ,{\mathbf{x}}_i^K\}$$X_i=\left\{{\mathbf{x}}_i^1,{\mathbf{x}}_i^2,\dots ,{\mathbf{x}}_i^K\right\}$X_(i)={x_(i)^(1),x_(i)^(2),dots,x_(i)^(K)}X_{i}=\left\{\mathbf{x}_{i}^{1}, \mathbf{x}_{i}^{2}, \ldots, \mathbf{x}_{i}^{K}\right\} with ${\mathbf{x}}_i^j\in {\mathbb{R}}^N$${\mathbf{x}}_i^j\in {\mathbb{R}}^N$x_(i)^(j)inR^(N)\mathbf{x}_{i}^{j} \in \mathbb{R}^{N}, while $y_i$$y_i$y_(i)y_{i} refers to the entire bag $X_i$$X_i$X_(i)X_{i} and not to any one instance ${\mathbf{x}}_i^j$${\mathbf{x}}_i^j$x_(i)^(j)\mathbf{x}_{i}^{j}. The goal in this scenario is to learn a bag classifier. Since a bag is an unordered set of instances without dependencies between its members, our classifier should be permutation-invariant with respect to the ordering of the bag instances. Theoretical results [43] suggest that a bag function $f(X)$$f(X)$f(X)f(X) is permutation-invariant if and only if it can be decomposed in the form: 

where $\varphi$$\varphi$phi\phi is an embedding function ${\mathbb{R}}^N\mapsto {\mathbb{R}}^M,\mathbf{z}\in {\mathbb{R}}^M$${\mathbb{R}}^N\mapsto {\mathbb{R}}^M,\mathbf{z}\in {\mathbb{R}}^M$R^(N)|->R^(M),zinR^(M)\mathbb{R}^{N} \mapsto \mathbb{R}^{M}, \mathbf{z} \in \mathbb{R}^{M} is the embedding of $X$$X$XX and $\rho$$\rho$rho\rho a suitable transformation ${\mathbb{R}}^M\mapsto \mathcal{Y}$${\mathbb{R}}^M\mapsto \mathcal{Y}$R^(M)|->Y\mathbb{R}^{M} \mapsto \mathcal{Y} (with $\mathcal{Y}$$\mathcal{Y}$Y\mathcal{Y} we denote the label domain). 

For models based on neural networks, the transformation $\varphi$$\varphi$phi\phi is usually a high-capacity CNN that is used either for feature extraction or for direct instance classification. The transformation $\rho$$\rho$rho\rho is then either a classification head that takes us from the embedding space to the class space, or simply the identity. A rather interesting modification to the above stems from incorporating an attention mechanism on the sum of Equation 3. This approach [44] defines the bag embedding as a non-linear combination (with learnable parameters $\mathbf{V},\mathbf{w}$$\mathbf{V},\mathbf{w}$V,w\mathbf{V}, \mathbf{w} ) of the instance features: 

The attention parameters $\mathbf{V}$$\mathbf{V}$V\mathbf{V} and $\mathbf{w}$$\mathbf{w}$w\mathbf{w} can be easily modelled as neural networks, thus allowing the whole model to be learnable end-to-end. In addition, attention scores provide an elegant way of identifying key instances within a bag. Attention-based MIL has been successfully applied to many problems [22], [45], [46]. Owing to its attractive properties and performance, we will use it as our core MIL model, which we will enhance in the next section with a semi-supervised component. 

In this section, we present our approach for utilizing unlabelled bags of instances in order to improve a multipleinstance classifier. As VAT provides a principled and elegant way of using unlabelled data, we elect to use it for semisupervised MIL, over other SSL approaches (e.g. Mean Teachers) that could be directly applied to the same problem. 
First, we extend VAT to the multiple-instance scenario. To that end, we introduce the concept of bag perturbation which is a set $R=({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_K)$$R=\left({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_K\right)$R=(r_(1),r_(2),dots,r_(K))R=\left(\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots, \mathbf{r}_{K}\right) that when added elementwise to a given bag $X$$X$XX, slightly perturbs it. The Multiple-Instance Local Distributional Smoothing (MI-LDS) loss can now be defined as: